Prediction of Energy Storage Performance in Polymer Composites Using High‐Throughput Stochastic Breakdown Simulation and Machine Learning

Abstract Polymer dielectric capacitors are widely utilized in pulse power devices owing to their high power density. Because of the low dielectric constants of pure polymers, inorganic fillers are needed to improve their properties. The size and dielectric properties of fillers will affect the dielectric breakdown of polymer‐based composites. However, the effect of fillers on breakdown strength cannot be completely obtained through experiments alone. In this paper, three of the most important variables affecting the breakdown strength of polymer‐based composites are considered: the filler dielectric constants, filler sizes, and filler contents. High‐throughput stochastic breakdown simulation is performed on 504 groups of data, and the simulation results are used as the machine learning database to obtain the breakdown strength prediction of polymer‐based composites. Combined with the classical dielectric prediction formula, the energy storage density prediction of polymer‐based composites is obtained. The accuracy of the prediction is verified by the directional experiments, including dielectric constant and breakdown strength. This work provides insight into the design and fabrication of polymer‐based composites with high energy density for capacitive energy storage applications.


Stochastic breakdown Model
In the stochastic breakdown model, the breakdown probability P(r) is written as (1) where E(r) is the electric field of a local point determined by the externally applied voltage and the microstructure, and E b (r) is the corresponding intrinsic breakdown strength determined by the polymer-based composites, and the summation in the denominator is the sum over all points that the local electric field exceeds the breakdown strength.
The local electric field distribution is obtained by solving the electrostatic equilibrium equation using an spectral iterative perturbation method. [1] As we assume there exist no spontaneous polarization in the system and the relative permittivity is independent of the applied field strength, the electric displacement D(r,E) can be expressed using Einstein notation as (2) where k(r) is the relative dielectric constant of the composite material.
According to Gauss's law the gradient of the electric displacement equals the positiondependent free charge density, i.e.
We know that total electric field equals to external field, E ext , plus the depolarization where φ(r) is the "depolarization potential". Using this substitution, Eq. 3 becomes We can express the inhomogeneous relative dielectric constant as the sum of the homogeneous reference k 0 ij and a perturbation Δk ij (r), i.e.
After substituting Eq. 6 into Eq. 5 and rearranging we obtain (7) From which we can solve for the depolarization potential and thus the electric field.
Details of the numerical method for solving this equation are described. [1] A grid size of N x Δx×N y Δx×N z Δx is employed in all simulations. For all 2D simulations, N x = N y = 256, N z = 1, and for all 3D simulations N x = N y = N z = 128. As an example, the PI/Al 2 O 3 composites is considered with isotropic relative permittivity of ε F = 10 and ε M = 3.5 for Al 2 O 3 and PI, respectively. The relative permittivity of the breakdown phase ε B is considered to be isotropic and to have a value of 10 4 to reflect its strong polarized ability due to the abundant space charge in the breakdown region. The breakdown strength of PI is 280 kV/mm through testing. Due to the lack of experimental data on the intrinsic breakdown strength of Al 2 O 3 filler, a value of 800 kV/mm is assigned for it. For each polymer-based composite, all fillers are generated with random distribution without overlapping.             Machine learning Table S2. The coefficient of determination adjusted-R 2 in 1st round of least square regressions with only one variable and one of the 12 prototypical functions.
Descriptor  Table S3. The coefficient of determination adjusted-R 2 in 2nd round of least square regressions with two and three variables and corresponding best prototypical functions selected from Supporting Information Table S2, where x 1 takes the top three best prototypical functions, x 2 and x 3 take the best prototypical function.
Descriptor Adjusted- Descriptor Adjusted- Descriptor Adjusted-   Where P is used as an inspection standard in regression analysis, for indicates that its influence on the dependent variable has no statistical significance when other variables remain unchanged, because the P value is greater than 0.05. Finally, the expression given by: was obtained from Aladdin.

Preparation of PEI/Al 2 O 3 composites
The PEI/Al 2 O 3 composites loaded with different filler contents were fabricated by the physical blending method and the hot-pressing method as shown in Supporting Information

Directional experimental simulation and testing
The Porod profiles of PEI/Al 2 O 3 composites are shown in Supporting Information Figure   S17. It can be seen that the curve has a certain degree of negative deviation. It is believed that the interaction between polymer molecular chains and Al 2 O 3 fillers is responsible for the negative deviation in Porod plots. From the slope, with the increase of v, the curve of PEI/Al 2 O 3 composites almost changes little, which again proves that the matrix and filler compatibility of each PEI/Al 2 O 3 composite is good. Supporting Information Figure S18 is the SAXS diagram of the PEI/Al 2 O 3 composites after Lorentz correction, which is used to detect the long cycle of semicrystalline polymer, generally appearing in the curve at the peak (highest point). However, it can be seen from Supporting Information Figure S18 that the curve keeps rising without a peak, indicating that the polymer is an amorphous polymer and that the fillers did not change the crystal state of the polymer.   kV/mm in these results, so it is necessary to calculate the breakdown value using the twoparameter Weibull distribution function for the simulation results.